Tutita M. Casa, Kyle Evans, Janine M. Firmender and Madelyn W. Colonnese

When teachers understand the key purposes for having students do mathematical writing, they can use that writing to deepen understanding.

It's time for math class at Valley Elementary, and today in Room 24, Mrs. Sanchez asks her class to write in their journals. Students walk to the neat, color-coded bins to collect their personalized journals, and then turn their attention to describing the similarities and differences between the attributes of squares and rectangles. Down the hall, Mr. Benson invites his students to write to defend why larger denominators make for smaller fractions—or why they don't. Upstairs, Ms. Park's students write their math autobiographies, and Mrs. Bell has students journal about how the Egyptian numeration system is different from the one people in western cultures use today. Across the hall, some of Mr. Rossi's students jot down initial thoughts about the difference between the "adding to" and "taking from" subtraction strategies.

Valley Elementary is a hypothetical school we use here to highlight how teachers can implement different types of mathematical writing for distinct purposes. Although many mathematics teachers have attended to calls by major policy groups to have students write during math (National Council of Teachers of Mathematics, 2000, 2014; National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010), until now, it's been unclear to many why it's a good idea to do so. A closer look at the kind of writing taking place in elementary schools illuminates the various purposes for engaging students in diverse kinds of writing exercises—exercises that result in different types of writing connected to math.

"Writing About Mathematics" Versus "Mathematical Writing"

Teachers have received limited guidance about writing during math instruction. To improve this situation, in 2015, the Elementary Mathematical Writing Task Force, of which three of us were members (Casa et al., 2016), convened to clarify different reasons why students should write in math class and to provide recommendations for writing that might further students' mathematics learning.^{1}
To ensure that its eventual recommendations would be appropriate to use with all students, this task force included teachers, math coaches, and university faculty with expertise in mathematics education, writing education, early childhood education, special education, English language learning, and gifted education.

The task force sorted through different writing activities that typically take place in math classrooms. One category, they realized, encompasses activities in which students write about mathematics, which prioritize the learning of literacy over mathematics. Examples of this type of writing include asking students to write about the lives of mathematicians or the math autobiography Ms. Park assigned.

The task force highlighted a second category—mathematical writing—that seems to reflect the National Commission on Writing's (2003) position that "at its best, writing is learning" (p. 51). At its heart, mathematical writing forefronts mathematical reasoning over literacy skills. This writing involves text; it can also include symbols (like the equal sign), numerals, operations, and such visual representations as pictures, charts, or tables unique to the discipline of mathematics.

Ultimately, the task force identified four types of mathematical writing that serve the overarching goals of engaging students in mathematical reasoning and communication: exploratory, informative/explanatory, argumentative, and mathematically creative writing. Let's consider these four types of mathematical writing and the purposes of each more closely.

1. Exploratory Writing: Making Sense of It

In Mr. Rossi's class, students were identifying the differences between two subtraction strategies. Writing was one strategy available for exploring this question, and Mr. Rossi reminded his students that they could write to work out their solutions and confusions in their math journals if they were so inclined. Some students decided to chat with a partner, others used manipulatives, and a few jotted down their initial ideas or worked out some specific subtraction examples.

The students choosing to write were engaged in exploratory writing—writing that helps students make sense of a problem or situation and sort through their own thoughts about mathematical concepts. Such writing can be initiated by the student at any time. It may be used more frequently when beginning a mathematical task as a way to brainstorm a problem's possible solution(s), ask questions, or work out confusions. The organic nature of this writing means that it's unlikely any two students' papers will look alike. Figure 1 shows a sample of a student's exploratory writing. This student wrote notes to herself at the top of the paper before answering the question about visual representations of fractions.^{2}

Figure 1. A 4th Grader's Exploratory Writing

2. Informative/Explanatory Writing: Clarifying to Others

Activities like Mrs. Sanchez's request to write about the similarities and differences between a square and a rectangle are examples of informative/explanatory writing, which positions students to describe and explain mathematical ideas. This kind of writing is a good opportunity for teachers to remind students that they should be clear in their writing and to guide students so their written messages are mathematically accurate.

Students can be asked to write descriptions of mathematical concepts, representations, and definitions, among other tasks. They may also be invited to provide mathematical explanations; besides explanations for solving a problem, these might include writing about mathematical connections and making comparisons between different representations or comparisons to real-world applications of math. Students often use this type of writing to communicate to others, such as their teachers or peers. Figure 2 displays a kindergartener's explanation of one way to measure the height of a drinking cup.

Figure 2. A Kindergartener's Informative/Explanatory Writing

3. Argumentative Writing: Making Your Case

Remember the students in Mr. Benson's class writing about why or why not larger denominators make for smaller fractions? They were exhibiting argumentative writing in which learners "construct viable arguments and critique the arguments of others" (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 6). It's likely that a few of these students would have written that "I just know it," and would need to be reminded that they should clearly state their position and back it with evidence. In addition to justifying their own positions, students read, analyze, and evaluate the mathematical arguments of others, taking the opportunity to either strengthen a peer's argument or disagree with it while providing their own evidence or counterexamples. For instance, Figure 3 shows a student's argument to support his claim that one measuring tool is more effective than others for a specific task.

Although the objective of Mrs. Bell's lesson was to find similarities between the Egyptian numeration system and the one we use today, one student instead invented her own numeration system, which she recorded in her journal.

In another case, a 4th grader whose teacher showed her work to members of the task force as an example of innovative thinking in math, initially made a table that was organized similarly to the strategy discussed in class for solving a rich task. She then made a second table organized differently, highlighted the pattern, and described her own formula for solving the problem. She explained how she arrived at the correct answer every time using each approach.

These students were engaging in mathematically creative writing, in which students think creatively and document mathematical ideas that extend beyond the intended outcome or process of solving a problem. This includes students generating original ideas, posing novel problems or questions, and displaying flexibility and fluency in ideas. Writing down outside-the-box ideas in this manner is something mathematicians do regularly.

To be considered original, the resulting ideas need not be original to the whole field of mathematics, but rather original to the student posing the idea—or to the class. Flexibility and fluency in ideas indicate that students are able to approach problems in unique ways and generate multiple ideas for the same problem. Mathematically creative writing also includes students elaborating on their ideas by writing extensions of their understanding or creating or developing mathematical generalizations.

Key Questions to Help You Make Instructional Choices

If you're a math teacher, you may have already seen students engaging in the kinds of writing reflected in the examples described here. If so, being clear about these four types of mathematical writing might help inform your decisions about incorporating more writing into your classes. To make sure any assigned or suggested writing has the greatest impact possible on students' math learning, it's important to consider the specific purposes characteristic of each type of mathematical writing and to address these purposes in your teaching.

The Elementary Mathematical Writing Task Force also pondered four questions related to the logistics of writing in math. We share here several of the Task Force's suggestions that might guide you in using writing to fuel math learning.

Should all students write mathematically?

Teachers may wonder about the appropriateness of having very young students and students with different learning needs or difficulties do mathematical writing. The task force considered this question and believes their recommendations for mathematical writing can be implemented across the elementary grades, starting in kindergarten. The sophistication of the writing, of course, will develop across grades and throughout experiences within a school year. These recommendations apply to all types of learners, including English language learners and youth identified as gifted. Educators need to provide students with good supports to engage in mathematical writing, such as encouraging students to write down initial ideas, allowing them to write in their native language, or modeling for students how to write down thoughts they have expressed orally.

How much time will guiding students to do this take?

Learning to write mathematically takes time—but it's time well spent. Imagine yourself teaching a child how to swim. Just as you wouldn't begin by leaving the child in the water and yelling, "Don't drown!" you wouldn't assign students to construct a complete mathematical argument or tackle similarly complex math writing tasks without giving them appropriate scaffolding and time. Teachers can share exemplary models of mathematical writing, have students discuss how to improve samples of work in progress, or provide sentence starters ("I agree with ___ because ___").

Ultimately, a child can only learn to swim by actually being in the water; the same could be said of mathematical writing. That means educators should give time and attention to consistently and meaningfully incorporating writing throughout the school year, in all units, and ultimately across grades. Drawing from our experiences, we suggest that elementary students focus on writing mathematically roughly every three or four days. This strikes a nice balance between students grappling with mathematical content in other ways, such as talking about a rich problem, and focusing on more individualized writing exercise (Casa, 2014).

Does it matter who the audience is?

Prompts for math writing often don't make clear to students who the audience is, and the teacher acts as the default. Students may not be as explicit or precise in their writing if they think that their teachers already know what they intend to say. If we tell students they are writing for their peers, they may be more inclined to ensure that the readers understand their points. The tone, clarity, and organization of their writing might be even more polished when the audience is more authentic and includes people outside their classroom, such as other teachers, parents, or even community leaders.

What forms can mathematical writing take?

The task force identified various forms—such as essays, word problems, concept maps, and math dictionary entries—that students can use to communicate in their writing. When trying to decide what form you might have students use, remember the key distinctions between mathematical writing (emphasizing mathematical reasoning) and writing about math (focusing on literacy). Keep in mind that the form writing takes is secondary to the types and purposes of the mathematical writing.

An Underused Tool

Mathematical writing is a tool that can further students' reasoning and communication. We hope that clarifying four powerful types of mathematical writing (and appropriate purposes for each), giving guidance on planning and implementing mathematical writing activities, and providing some samples of good student math writing will give math educators insights into what—until now—has been an underutilized tool.

Authors' Note: This material is based on work supported by the National Science Foundation under Grant No. 1545908, Task Force on Conceptualizing Elementary Mathematical Writing: Implications for Mathematics Education Stakeholders. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

References

Casa, T. M. (2014). Supporting writing with the student mathematician discourse framework. In K. Karp (Ed.), Annual perspectives in mathematics education 2014: Using research to improve instruction (pp. 107–117). Reston, VA: National Council of Teachers of Mathematics.

Casa, T. M., Firmender, J. M., Cahill, J., Cardetti, F., Choppin, J. M., Cohen, J. … Zawodniak, R. (2016). Types of and purposes for elementary mathematical writing: Task force recommendations. Retrieved from http://mathwriting.education.uconn.edu.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors. Retrieved from www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

^{2}
All figures are from Casa, T. M., Firmender, J. M., Cahill, J., Cardetti, F., Choppin, J. M., Cohen, J. … Zawodniak, R. (2016). Types of and purposes for elementary mathematical writing: Task force recommendations.

Tutita M. Casa is assistant professor in the Department of Curriculum and Instruction in the Neag School of Education and Kyle Evans is a doctoral candidate in mathematics at the University of Connecticut, Storrs. Janine M. Firmender is assistant professor in the Department of Teacher Education at St. Joseph's University in Philadelphia. Madelyn W. Colonnese is a doctoral candidate in elementary mathematics education at the University of Connecticut, Storrs.